Relativized Worlds with an Innite Hierarchy Lance Fortnow y University of Chicago Department of Computer Science 1100 E. 58th. St. Chicago, IL 60637 Abstract We introduce the \Book Trick" as a method for using results about random oracles to create relativized worlds with an innite polynomial-time hierarchy. We use it to show, for example, a single relativized world where One can nd satisfying assignments to satisable formulae with only nonadaptive queries to an NP oracle, and the counting class SPP strictly contains the polynomial-time hierarchy, and the polynomial-time hierarchy is innite. Keywords: Theory of Computation; Computational Complexity; Relativization; Polynomial-Time Hierarchy; Ronald Book 1 Introduction Creating a relativized world where the polynomial-time hierarchy is innite was one of the more dicult constructions in the mid-1980's. The eld of circuit complexity was invented initially to tackle this problem (see [FSS84]). The rst such construction by Yao [Yao85] was a technical masterpiece. These days we often have the task of making some other proposition true while making the polynomial-time hierarchy innite. Consider, for example, the following proposition. Proposition 1.1 One can nd satisfying assignments to satisable formulae using nonadaptive queries to an NP oracle. Finding satisfying assignments using an NP oracle is an easy process using the self-reducing properties of SAT. However this algorithm requires adaptive queries to SAT. One might conjecture that Proposition 1.1 implies the polynomial-time hierarchy collapses. We would like to give a relativized counterexample to this conjecture, i.e., create a relativized world where Proposition 1.1 holds and the polynomial-time hierarchy is innite. This paper is dedicated to the memory of Ronald Book. May his love of structural complexity theory and his desire to share it with the world be an inspiration to us all. y URL: http://www.cs.uchicago.edu/~fortnow. Email: fortnow@cs.uchicago.edu. Supported in part by NSF grant CCR 92-53582. 1
Creating such worlds is still a dicult task. One can try a direct construction, using such combinatorial techniques such as the Hastad switching lemma [Has89] but these approaches are complicated at best. Another useful but limited direction involves generic oracles (see [FFKL93]). In this paper, we introduce a new technique using random oracles. We show how to use a recent result by Ron Book to turn results on random oracles to results on oracles that make the polynomial-time hierarchy innite. Ron Book [Boo94] showed that if the polynomial-hierarchy collapses relative to a random oracle then it collapses unrelativized. We use this result to formulate a technique we call the Book Trick: Any relativizing proof that a proposition holds relative to a random oracle immediately implies that there exists an oracle that makes the proposition true and simultaneously makes the polynomial-time hierarchy innite. Watanabe and Toda [WT93] give a relativizable proof that Proposition 1.1 holds relative to a random oracle. Using our techniques, we immediately get that there exists a relativized world where the polynomial-time hierarchy is innite and Proposition 1.1 holds. We also examine the counting class SPP dened by Fenner, Fortnow and Kurtz [FFK94] and give a relativized world where SPP strictly contains an innite polynomial-time hierarchy. 2 Preliminaries We assume the reader is familiar with Turing machines and standard complexity classes such as P, NP and conp and the concept of relativization. Fortnow [For94] gives an overview of relativization and its importance in complexity theory. The polynomial-time hierarchy is a generalization of NP dened inductively. We let p 0 = P and p i+1 = NP p i for i 0. We let PH = S i0 p i. We say the polynomial-time hierarchy collapses if p i = p i+1 for some i, otherwise we say the polynomial-time hierarchy is innite. Most complexity theorists conjecture that the polynomial-time hierarchy is innite. We let L (x) be the characteristic function of L, i.e., L (x) = 1 if x is in L and L (x) = 0 otherwise. Let hx; yi be a standard polynomial-time computable and invertible tupling function. We use A B to represent the join of A and B, i.e., A B = fhx; ii j (i = 0 and x 2 A) or (i = 1 and x 2 B)g: We use AB to be the disjoint union of A and B, i.e. AB = (A? B) [ (B? A). 3 The Book Trick Creating relativized worlds where the polynomial-time hierarchy is innite is a dicult process requiring combinatorial bounds on bounded depth circuits. The rst such oracle is due to Yao [Yao85]. Theorem 3.1 (Yao) There exists an oracle A such that for all i 0, A i 6= A i+1. His proof has been simplied and improved by Hastad [Has89]. 2
Because of the dicultly of the construction, complexity theorists have had little luck creating relativized worlds where the polynomial-time hierarchy is innite and other specic complexity statements hold. One notable exception is the work of Sheu and Long [SL96] separating the low and high hierarchies. We show how to use the theory of random oracles to help create relativized worlds where the polynomial-time hierarchy is innite. Bennet and Gill [BG81] rst looked at random oracles. Denition 3.2 (Bennet-Gill) A relativizable proposition P holds relative to a random oracle if Pr Rf0;1g (P R ) = 1: Bennet and Gill show that P 6= NP relative to a random oracle. Many complexity theorists had believed in the \random oracle hypothesis" where propositions holding relative to a random oracle should also hold in the unrelativized world. Recent work on interactive proofs have put that hypothesis to rest (see [CCG + 94]). Still we can use tools for random oracles to show that certain relativized worlds exist. One of these tools is the Kolmogorov Zero-One law. Lemma 3.3 If P is a relativizable proposition such that P A, P AB for all oracles A and nite sets B then Pr (P R ) 2 f0; 1g: Rf0;1g Since we can encode nite dierences of an oracle in the code of a Turing machine, nearly all propositions P have the property needed for Lemma 3.3. Ron Book [Boo94] proves the following result. Theorem 3.4 (Book) If relative to random R the polynomial-time hierarchy collapses then the (unrelativized) polynomial-time hierarchy collapses. A close examination of Book's proof and the proof of Nisan and Wigderson [NW94] that Book uses shows that Theorem 3.4 relativizes. Also for our purposes, the contrapositive will be a more useful form for us. This yields the following corollary. Corollary 3.5 (Book) For every oracle A, if the polynomial-time hierarchy is innite relative to A then for random R, the polynomial-time hierarchy relative to A R is innite. From Corollary 3.5 we can then turn relativizable theorems about random oracles into results about an innite polynomial-time hierarchy. Lemma 3.6 (The Book Trick) If P is a relativizable proposition such that for all A and random R, P AR is true then there exists an oracle B such that P B holds and the polynomial-time hierarchy is innite relative to B. Proof: By Theorem 3.1, let A be an oracle relative to which the polynomial-time hierarchy is innite. Then for random R we have P AR and the polynomial-time hierarchy is innite relative to A R. Fix such an R and let B = A R. 2 Note that Lemma 3.6 does not imply that the polynomial-time hierarchy is innite relative to a random oracle. This remains an important unproven conjecture. We can immediately apply Lemma 3.6 to give us many interesting relativized worlds. 3
Corollary 3.7 There exists a single relativized world where NP does not have measure zero in EXP, One can nd a satisfying assignment of a satisable formula with nonadaptive queries to NP, P = BPP, and the polynomial-time hierarchy is innite. The denitions and motivation of measure within EXP is beyond the scope of this paper. We recommend the survey by Lutz [Lut97]. Buhrman and Thierauf [BT96] give the rst report of an oracle relative to which one can nd a satisfying assignment of a satisable formula with nonadaptive queries to NP and the polynomialtime hierarchy is innite. We give the rst proof in this paper. Proof of Corollary 3.7: Kautz and Milterson [KM96], Watanabe and Toda [WT93] and Bennet and Gill [BG81] respectively show that the rst three statements hold relative to a random oracle and all of their proofs relativize. Corollary 3.7 immediately follows from Lemma 3.6. 2 We can extend Corollary 3.7 to most statements known to hold relativize to a random oracle such as NP = AM [NW94], the failure of the Berman-Hartmanis isomorphism conjecture [KMR95], the existance of extremely hard one-way functions [IR89, KMR95] and the existance of disjoint pairs of NP (or conp) sets that are not P-separable [Ver93]. 4 The Complexity of SPP We can also use the Book Trick to show the potential surprising power of the counting class SPP. A function f is a GapP function if there exists a polynomial-time nondeterministic Turing machine M such that for all x, f(x) is the dierence of the number of accepting computation paths of M(x) and the number of rejecting computation paths of M(x). Fenner, Fortnow and Kurtz [FFK94] dene the GapP functions and show many interesting properties of them. They also dene the counting class SPP. A language L is in SPP if the characteristic function of L is a GapP function. Fenner, Fortnow and Kurtz [FFK94] show that in a formal sense SPP is the smallest possible nontrivial Gap-denable counting class. Han, Hemaspaandra and Thierauf [HHT97] examine a class BPP path showing BPP path is contained in the polynomial-time hierarchy, and There is a relativized world where SPP is not contained in BPP path. A natural question arises as to whether one can simply create a relativized world where SPP does not lie in the polynomial-time hierarchy. Fortnow [For97], in his survey on counting complexity, states the following result without proof. Theorem 4.1 There exists a relativized world where SPP strictly contains an innite polynomialtime hierarchy. 4
Theorem 4.1 gives an interesting contrast with the following result proven by Fenner, Fortnow, Kurtz and Li [FFKL93] using generic oracles. Theorem 4.2 There exists a relativized world where P = SPP and the polynomial-time hierarchy is innite. We present the rst published proof of Theorem 4.1. Proof of Theorem 4.1: Toda and Ogihara [TO92] prove the following interesting relationship between the polynomial-time hierarchy and GapP functions. Theorem 4.3 (Toda-Ogihara) Let f be in GapP PH and p be a polynomial. There exist a twoargument function g in GapP and a polynomial q such that for all x, Pr r2 q(n)(g(x; r) = f(x)) 1? 2?p(n) : From Theorem 4.3 we can easily show that SPP contains PH relative to a random oracle. Fix a language L in PH. Let f be the characteristic function for L and notice that f 2 GapP PH. Fix p(n) = 2n + 2 and let g and q be the results of applying Theorem 4.3. Let M be the polynomial-time nondeterministic Turing machine that denes the function g. Create a new relativized machine N R that works as follows: N R (x) gets r from the oracle by letting r i = R (hx; ii). The machine will then simulate M(x; r). Let h R (x) be the GapP function dened by N R. For any x, the probability over all R that N R (x) 6= L (x) is at most 2?2jxj+2. Summing this error over all strings gives that L is in SPP R via f with probability at least 1=2. By the Kolmogorov zero-one law (Lemma 3.3), we have that L is in SPP R for random R. The above argument relativizes so the Book Trick (Lemma 3.6) gives us a relativized world where SPP contains the polynomial-time hierarchy and the hierarchy is innite. We would then like to apply the following well-known lemma to show strict containment. Lemma 4.4 If there exists a set A in the polynomial-time hierarchy such that for every language L in the polynomial-time hierarchy, L polynomial-time many-one reduces to A then the polynomialtime hierarchy collapses. Proof: The set A must lie in some p i. Let L be a p i+1-complete set and we then have p i = p i+1. 2 If SPP R had a complete set A then by Lemma 4.4, A would not be in the polynomial-time hierarchy and we would have strict containment. However, we cannot assume SPP R has a complete set, in fact there are relativized worlds where SPP does not have complete sets (see [FFKL93]). We get around this problem by analyzing Toda and Ogihara's construction [TO92] for Theorem 4.3. If L R i is the standard relativizable complete set for p i then the construction produces the machine N above using running time O(n 10i ). We can then dene the relativized set A R by A R = fhi; 1 jxj10i ; xi j x 2 L R i g: A simple variation of the above argument shows that A R 2 SPP R for random R. Also, we have every language in PH R reduces to A so by Lemma 4.4 we have that A R 2 SPP R?PH R for random R. 2 5
5 Further Directions While questions about whether statements hold relative to a random oracle are no longer inherently interesting, they do give us a \probabilistic method" approach to showing the existence of an oracle making a proposition true. Also, by countable additivity of measure zero sets, all statements that hold relative to a random oracle, hold simultaneously relative to a random oracle. Our paper gives another purpose to studying random oracles: It gives us a method to show that the polynomial-time hierarchy is innite with respect to many other propositions. To this end, it is still worthwhile to study the truth of certain statements relative to random oracles, such as 1. Is the polynomial-time hierarchy innite? 2. Is P 6= NP \ conp? 3. Is P 6= BQP where BQP is the class of languages accepted eciently by quantum Turing machines? Showing the negation of the above would yield surprising results, the rst would collapse the hierarchy (Theorem 3.4) and either of the other two would yield a fast probabilistic algorithm for factoring. Finally, one should realize the limited interpretations of oracle results, particularly for results on interactive proof systems. One can use the Book Trick combined with work of Chang, Chor, Goldreich, Hartmanis, Hastad, Ranjan and Rohatgi [CCG + 94] to get a relativized world where the polynomial-time hierarchy is innite and some conp languages do not have interactive proofs. However, we know that this result does not hold in the unrelativized world [LFKN92]. Acknowledgments We thank Harry Buhrman, Lane Hemaspaandra, Thomas Thierauf and Alan Selman for discussions on the Book Trick. We also thank Hemaspaandra for encouraging the author to write this article. Finally we thank the anonymous referees for several suggested improvements on the exposition including the statement of the Book Trick in the introduction. References [BG81] C. Bennett and J. Gill. Relative to a random oracle, P A 6= NP A 6= co? NP A with probability one. SIAM Journal on Computing, 10:96{113, 1981. [Boo94] [BT96] R. Book. On collapsing the polynomial-time hierarchy. Information Processing Letters, 52(5):235{237, 1994. H. Buhrman and T. Thierauf. The complexity of generating and checking proofs of membership. In Proceedings of the 13th Symposium on Theoretical Aspects of Computer Science, volume 1046 of Lecture Notes in Computer Science, pages 75{86. Springer, Berlin, 1996. 6
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